Minimum covering reciprocal distance signless Laplacian energy of graphs

Let G be a simple connected graph. The reciprocal transmission Tr′_G(ν) of a vertex ν is defined as

TrG′(ν)=∑u∈V(G)1dG(u,ν), u≠ν.$${\rm{Tr}}_{\rm{G}}^\prime ({\rm{\nu }}) = \sum\limits_{{\rm{u}} \in {\rm{V}}(G)} {{1 \over {{{\rm{d}}_{\rm{G}}}(u,{\rm{\nu }})}}{\rm{u}} \ne {\rm{\nu }}.} $$

The reciprocal distance signless Laplacian (briefly RDSL) matrix of a connected graph G is defined as RQ(G)= diag(Tr′ (G)) + RD(G), where RD(G) is the Harary matrix (reciprocal distance matrix) of G and diag(Tr′ (G)) is the diagonal matrix of the vertex reciprocal transmissions in G. In this paper, we investigate the RDSL spectrum of some classes of graphs that are arisen from graph operations such as cartesian product, extended double cover product and InduBala product. We introduce minimum covering reciprocal distance signless Laplacian matrix (or briey MCRDSL matrix) of G as the square matrix of order n, RQ_C(G) := (q_i;j),

qij={1+Tr′(νi)ifi=jandνi∈CTr′(νi)ifi=jandνi∉C1d(νi,νj)otherwise$${{\rm{q}}_{{\rm{ij}}}} = \left\{ {\matrix{ {1 + {\rm{Tr}}\prime ({{\rm{\nu }}_{\rm{i}}})} & {{\rm{if}}} & {{\rm{i = j}}} & {{\rm{and}}} & {{{\rm{\nu }}_{\rm{i}}} \in {\rm{C}}} \cr {{\rm{Tr}}\prime ({{\rm{\nu }}_{\rm{i}}})} & {{\rm{if}}} & {{\rm{i = j}}} & {{\rm{and}}} & {{{\rm{\nu }}_{\rm{i}}} \notin {\rm{C}}} \cr {{1 \over {{\rm{d(}}{{\rm{\nu }}_{\rm{i}}},{{\rm{\nu }}_{\rm{j}}})}}} & {{\rm{otherwise}}} & {} & {} & {} \cr } } \right.$$

where C is a minimum vertex cover set of G. MCRDSL energy of a graph G is defined as sum of eigenvalues of RQ_C. Extremal graphs with respect to MCRDSL energy of graph are characterized. We also obtain some bounds on MCRDSL energy of a graph and MCRDSL spectral radius of 𝒢, which is the largest eigenvalue of the matrix RQ_C (G) of graphs.